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We demonstrate the presence of a resonant interaction between a pair of coupled quantum wires, which are formed in the ultrahigh mobility two-dimensional electron gas of a GaAs/AlGaAs quantum well. The coupled-wire system is realized by an extension of the split-gate technique, in which bias voltages are applied to Schottky gates on the semiconductor surface, to vary the width of the two quantum wires, as well as the strength of the coupling between them. The key observation of interest here is one in which the gate voltages used to define one of the wires are first fixed, after which the conductance of this wire is measured as the gate voltage used to form the other wire is swept. Over the range of gate voltage where the swept wire pinches off, we observe a resonant peak in the conductance of the fixed wire that is correlated precisely to this pinchoff condition. In this paper, we present new results on the current- and temperature-dependence of this conductance resonance, which we suggest is related to the formation of a local moment in the swept wire as its conductance is reduced below 2e2/h.